Question

+ (20) 3. Indicate whether the reasoning of each of the following statements is correct or incorrect. Explain why or why not in each case. (Note: For an "if-then" statement, you do not need to verify that the hypothesis of the statement is true, nor come to any final conclusion ab f(x) is irreducible. Just indicate whether the conclusion correctly follows from the assumptions.) a) f(x) = +422 - 2x - 20 is irreducible in Qlx) by Eisenstein's criterion. b) If f(1), f(-1), (2), and f(-2) are all nonzero, then f(x) = 3 + 0-2 is irreducible in Q[3]. c) If f(x) = x + 2x2 + 4x+6 is irreducible in Q[-], then f(x) is also irreducible in 2 [-] d) If y(x) = x2 +21² +1+4 is irreducible in Z7[3], then f(x) = ??+9x2 - 63 - 10 is irreducible in QE). † (20) 4. Indicate whether each statement below must be true or might be false. Explain each answer brictly. (Two points for each question will be determined by the written explanation.) T F Every real number v is a root of some nonzero polynomial f(x) in Q[U]. T F Every element in Q() can be written as a + 2 for some a and bin Q. T F Every element in Q(V2,VT) can be written as a + b 2 + cv7 for some a, b, cc. T F Every irrational number in QCV) is a root of a decree three polynomial in Q T F . There is a field containing exactly 80 elementa MacBook Pro Q Search or enter website name 8

Question 3 please

Answer 1

O @ liet fra) = 5+4624-20 Then f(2) ů inreducible in grm, by Eisenstein's criterim. The statement is incorrect, because no such prime p exist such that PX1, P14, P/2 ,8/20 and 8X20 © 97411), 4(-1), H(2) and f(2) are all non zero, then f(x) = ñ + 23 -2 ů irreducible in gen] The statement is incorrect gf f(x) = n3+25+un+6 is irreducible in Q [m], then fin is also irreducible in Zz (m) The statement is incorrect in the conclusion correctly does not follows from the ap cumptim. @ of g(x) = % + 25 th th iů inneducible in Zg m), then finanqr6n -10 år inneducible in Q13) Here the statement ů correct ie the conclusion correctly follown from the accumption.